The Real K -Theory of Compact Lie Groups
Chi-Kwong FOK
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA E-mail: ckfok@math.cornell.edu
URL: http://www.math.cornell.edu/~ckfok/
Received August 22, 2013, in final form March 06, 2014; Published online March 11, 2014 http://dx.doi.org/10.3842/SIGMA.2014.022
Abstract. Let G be a compact, connected, and simply-connected Lie group, equipped with a Lie group involution σG and viewed as a G-space with the conjugation action. In this paper, we present a description of the ring structure of the (equivariant) KR-theory of (G, σG) by drawing on previous results on the module structure of theKR-theory and the ring structure of the equivariantK-theory.
Key words: KR-theory; compact Lie groups; Real representations; Real equivariant forma- lity
2010 Mathematics Subject Classification: 19L47; 57T10
1 Introduction
The complex K-theory of compact connected Lie groups with torsion-free fundamental groups was worked out by Hodgkin in the 60s (cf. [13] and Theorem 2.4). It asserts that theK-theory ring is theZ2-graded exterior algebra over Z on the module of primitive elements, which are of degree −1 and associated with the representations of the Lie group. For the elegant proof of Hodgkin’s result in the special case whereG is simply-connected, see [1].
In [2], Atiyah introduced KR-theory, which is basically a version of topological K-theory for the category of Real spaces, i.e. topological spaces equipped with an involution. KR-theory can be regarded as a hybrid of KO-theory, complex K-theory and KSC-theory (cf. [2]). One can also consider equivariant KR-theory, where a certain compatibility condition between the group action and the involution is assumed. For definitions and some basic properties, see Definitions 2.26and 2.27[2,5].
Since Hodgkin’s work, there have appeared two kinds of generalizations ofK-theory of com- pact Lie groups. The first such isKR-theory of compact Lie groups, which was first studied by Seymour (cf. [15]). He obtained the KR∗(pt)-module structure of KR∗(G), whereGis a com- pact, connected and simply-connected Lie group equipped with a Lie group involution, using his structure theorem of KR-theory of a certain type of spaces (cf. Theorems2.32,2.37 and 2.40).
He was unable to obtain a complete description of the ring structure, however, and could only make some conjectures about it. In [7], Bousfield determined functorially the united 2-adic K- cohomology algebra of any compact, connected and simply-connected Lie group, which includes the 2-adicKO-cohomology algebra, and hence extended Seymour’s results in the 2-adic case, if the Lie group involution is taken into account appropriately.
The second one is the equivariant K-theory of compact Lie groups. In [9], Brylinski and Zhang showed that, for a compact connected Lie group Gwith torsion free fundamental group, the equivariantK-theory,KG∗(G), whereGacts on itself by conjugation, is isomorphic to the ring of Grothendieck differentials of the complex representation ring R(G) (for definition, see [9]).
It is noteworthy thatGsatisfies the property of being weakly equivariantly formal`a la Harada and Landweber (cf. Definition 4.1 of [11] and Remark 2.8).
In this paper, based on the previous works of Seymour’s and Brylinski–Zhang’s and putting both the Real and equivariant structures together, we obtain a description of the ring structure of the equivariantKR-theory of any compact, connected and simply-connected Lie group, which is recorded in Theorem4.33. We express the ring structure using relations of generators associated to Real representations ofGof real, complex and quaternionic type (with respect to the Lie group involution). Our main contribution is twofold. First, we observe that the conditions of Seymour’s structure theorem are an appropriate candidate for defining the notion of ‘Real formality’ in analogy to weakly equivariant formality. These notions together prompt us to introduce the definition of ‘Real equivariant formality’, which leads to a structure theorem for equivariant KR-theory (Theorem4.5). Any compact, connected and simply-connected Real Lie group falls under the category of Real equivariantly formal spaces and Theorem 4.5 enables us to obtain a preliminary description of the equivariant KR-theory as an algebra over the coefficient ring.
Second, inspired by Seymour’s conjecture, we obtain the squares of the real and quaternionic type generators, which in addition to other known relations complete the description of the ring structure. These squares are non-zero 2-torsions in general. Hence the equivariant KR-theory in general is not a ring of Grothendieck differentials, as in the case for equivariant K-theory.
Despite this, we remark that, in certain cases, if we invert 2 in the equivariant KR-theory ring, then the result is an exterior algebra over the localized coefficient ring of equivariantKR-theory.
The organization of this paper is as follows. In Section2, we review the (equivariant)K-theory of compact connected Lie groups with torsion-free fundamental groups, Real representation rings RR(G), KR-theory, and the main results in [15]. In Section 3 we give a description of the coefficient ring KR∗G(pt). Section 4 is devoted to proving Theorems 4.5 and 4.33, the main results of this paper, which give a full description of the ring structure of KR∗G(G). In Section5 we apply the main results to obtain the ring structure of the ordinaryKR-theory of compact Lie groups, thereby confirming Seymour’s conjecture on the squares of the real type generators and disproving that on the squares of the quaternionic type generators. We also work out several examples, one of which shows how equivariant KR-theory tells apart (while K-theory cannot) the ring structures of two equivariant KR-theory rings of G, where in one caseG acts on itself trivially, while in another Gacts by conjugation.
Throughout this paper, we follow the convention in [2] and [5] in reference toKR-theory. In particular, we use the word ‘Real’ (with a capital R) in all contexts involving involutions, so as to avoid confusion with the word ‘real’ with the usual meaning. For example, ‘Real K-theory’
is used interchangeably with KR-theory, whereas ‘realK-theory’ means KO-theory.
2 Preliminaries
Throughout this section, G denotes any compact Lie groups, and X, Y, . . . any finite CW- complexes unless otherwise specified.
2.1 (Equivariant) K-theory of compact connected Lie groups Recall that the functor K−1 is represented by U(∞) := lim
n→∞U(n), i.e. for any X, K−1(X) is the abelian group of homotopy classes of maps [X, U(∞)]. Such a description of K−1 leads to the following
Definition 2.1. Let δ :R(G) → K−1(G) be the group homomorphism which sends any com- plexG-representationρto the homotopy class ofi◦ρ, whereρ is regarded as a continuous map from GtoU(n) and i:U(n),→U(∞) is the standard inclusion.
In fact, any element inK−q(X) can be represented by a complex of vector bundles onX×Rq, which is exact outside X× {0} (cf. [3]). This gives another interpretation ofδ(ρ), as shown in the following proposition, which we find useful for our exposition.
Proposition 2.2. If V is the underlying complex vector space of ρ, then δ(ρ) is represented by 0→G×R×V →G×R×V →0,
(g, t, v) 7→ (g, t,−tρ(g)v) if t≥0, (g, t, v) 7→ (g, t, tv) if t≤0.
Proposition 2.3 ([13]).
1. δ is a derivation of R(G) taking values in K−1(G) regarded as an R(G)-module whose module structure is given by the augmentation map. In other words, δ is a group homo- morphism from R(G) to K−1(G) satisfying
δ(ρ1⊗ρ2) = dim(ρ1)δ(ρ2) + dim(ρ2)δ(ρ1). (2.1) 2. If I(G) is the augmentation ideal of R(G), then δ(I(G)2) = 0.
The main results of [13] are stated in the following
Theorem 2.4. Let G be a compact connected Lie group with torsion-free fundamental group.
Then
1. K∗(G) is torsion-free.
2. Let J(G) :=I(G)/I(G)2. Then the mapeδ :J(G)→K−1(G) induced by δ is well-defined, and K∗(G) =V
(Im(eδ)).
3. In particular, if G is compact, connected and simply-connected of rank l, then K∗(G) = V
Z(δ(ρ1), . . . , δ(ρl)), where ρ1, . . . , ρl are the fundamental representations.
Viewing G as a G-space via the adjoint action, one may consider the equivariant K-theo- ryKG∗(G). Let Ω∗R(G)/
Z be the ring of Grothendieck differentials ofR(G) overZ, i.e. the exterior algebra over R(G) of the module of K¨ahler differentials of R(G) overZ (cf. [9]).
Definition 2.5. Letϕ: Ω∗R(G)/
Z →KG∗(G) be theR(G)-algebra homomorphism defined by the following
1) ϕ(ρV) := [G×V]∈KG∗(G), where Gacts on G×V by g0·(g1, v) = (g0g1g0−1, ρV(g0)v), 2) ϕ(dρV) ∈ KG−1(G) is the complex of vector bundles in Definition 2.2 where G acts on
G×R×V by g0·(g1, t, v) = (g0g1g0−1, t, ρV(g0)v), We also defineδG:R(G)→KG−1(G) byδG(ρV) :=ϕ(dρV).
Remark 2.6. The definition of δG(ρV) given in [9], where the middle map of the complex of vector bundles is defined to be (g, t, v)7→(g, t, tρV(g)v) for all t∈R, is incorrect, asδG(ρV) so defined is actually 0 in KG−1(G). The definition given in Definition 2.5 is the correction made by Brylinski and relayed to us by one of the referees.
Theorem 2.7 ([9]).
1. δG is a derivation of R(G) taking values in the R(G)-moduleKG−1(G), i.e. δG satisfies δG(ρ1⊗ρ2) =ρ1·δG(ρ2) +ρ2·δG(ρ1). (2.2) 2. Let G be a compact connected Lie group with torsion-free fundamental group. Then ϕ is
an R(G)-algebra isomorphism.
Remark 2.8.
1. Although the definition ofδG(ρV) given by Brylinski–Zhang in [9] is incorrect, their proof of Theorem2.7can be easily corrected by using the correct definition as in Definition2.5, which does not affect the validity of the rest of their arguments, and Theorem 2.7 still stands.
2. In [11], aG-spaceXis defined to beweakly equivariantly formal if the mapKG∗(X)⊗R(G) Z → K∗(X) induced by the forgetful map is a ring isomorphism, where Z is viewed as an R(G)-module through the augmentation homomorphism. Theorem 2.4 and 2.7 imply that G is weakly equivariantly formal if it is connected with torsion-free fundamental group. We will make use of this property in computing the equivariant KR-theory of G.
3. Let f : KG∗(G) → K∗(G) be the forgetful map. Note that f(ϕ(ρ)) = dim(ρ) and f(ϕ(dρ)) = δ(ρ). Applying f to equation (2.2) in Theorem 2.7, we get equation (2.1) in Proposition2.3.
4. I(G), being a prime ideal in R(G), can be thought of as an element in SpecR(G), and K∗(G)∼=V
ZTI(G)∗ SpecR(G),KG∗(G)∼=V
R(G)TI(G)∗ SpecR(G).
2.2 Real representation rings
This section is an elaboration of the part on Real representation rings in [5] and [15]. Since the results in this subsection can be readily generalized from those results concerning the special case where σG is trivial, we omit most of the proofs and refer the reader to any standard text on representation theory of Lie groups, e.g. [6] and [8].
Definition 2.9. A Real Lie group is a pair (G, σG) where Gis a Lie group and σG a Lie group involution on it. A Real representation V of a Real Lie group (G, σG) is a finite-dimensional complex representation of G equipped with an anti-linear involution σV such that σV(gv) = σG(g)σV(v). LetRep
R(G, σG) be the category of Real representations of (G, σG). A morphism between V and W ∈ RepR(G, σG) is a linear transformation from V to W which commute withG and respectσV and σW. We denote Mor(V, W) by HomG(V, W)(σV,σW). An irreducible Real representation is an irreducible object in Rep
R(G, σG). The Real representation ring of (G, σG), denoted byRR(G, σG), is the Grothendieck group ofRepR(G, σG), with multiplication being tensor product over C. Sometimes we will omit the notation σG if there is no danger of confusion about the Lie group involution.
Remark 2.10. Let V be an irreducible Real representation of G. Then HomG(V, V)σV must be a real associative division algebra which, according to Frobenius’ theorem, is isomorphic to either R, C or H. Following the convention of [5], we call HomG(V, V)σV the commuting field of V.
Definition 2.11. If V is an irreducible Real representation of G, then we say V is of real, complex or quaternionic type according as the commuting field is isomorphic to R, C or H. Let RR(G,F) be the abelian group generated by the isomorphism classes of irreducible Real representations with Fas the commuting field.
Remark 2.12. RR(G)∼=RR(G,R)⊕RR(G,C)⊕RR(G,H) as abelian groups.
Definition 2.13. Let V be a G-representation. We use σG∗V to denote the G-representation with the same underlying vector space where the G-action is twisted by σG, i.e. ρσ∗
GV(g)v = ρV(σG(g))v. We will useσ∗G to denote the map onR(G) defined by [V]7→[σ∗GV].
Proposition 2.14. If V is a complex G-representation, and there exists f ∈ HomG(V, σG∗V) such thatf2 = IdV, thenV is a Real representation of Gwithf as the anti-linear involutionσV.
Proof . Iff ∈HomG(V, σG∗V), then it is anti-linear onV and f(gv) =σG(g)f(v) forg∈Gand v ∈ V. The assumption that f2 = IdV just says that f is an involution. So V together with
σV =f is a Real representation ofG.
Proposition 2.15. Let V be an irreducible complex representation of G and suppose that V ∼= σ∗GV. Let f ∈HomG(V, σG∗V). Then
1. f2 =kIdV for some k∈R.
2. There exists g∈HomG(V, σ∗GV) such thatg2= IdV or g2 =−IdV.
Proof . Note that f2 ∈ HomG(V, V). By Schur’s lemma, f2 = kIdV for some k ∈C. On the other hand,
f(kv) =f(f(f(v))) =kf(v).
But f is an anti-linear map on V. It follows that k = k and hence k ∈ R. For part (2), we may first simply pick an isomorphism f ∈HomG(V, σG∗V). Then f2 =kIdV for somek ∈R×. Schur’s lemma implies that anyg∈HomG(V, σ∗GV) must be of the formg=cf for somec∈C. Then g2 =cf ◦cf =ccf2 =|c|2kIdV. Consequently, if kis positive, we choose c= √1
k so that g2= IdV; ifkis negative, we choose c= √1−k so thatg2=−IdV. Proposition 2.16. Let V be an irreducible Real representation of G.
1. The commuting field ofV is isomorphic to RiffV is an irreducible complex representation and there exists f ∈HomG(V, σG∗V) such thatf2 = IdV.
2. The commuting field of V is isomorphic toC iffV ∼=W ⊕σG∗W as complexG-representa- tions, whereW is an irreducible complexG-representation andW σ∗GW, andσV(w1, w2)
= (w2, w1).
3. The commuting field of V is isomorphic to H iff V ∼= W ⊕σ∗GW as complex G-repre- sentations, where W is an irreducible complex G-representation and there exists f ∈ HomG(V, σG∗V) such that f2=−IdV, and σV(w1, w2) = (w2, w1).
Proof . One can easily establish the above proposition by modifying the proof of Proposition 3 in Appendix 2 of [6], which is a special case of the above proposition where σG is trivial.
Proposition 2.17.
1. The map i:RR(G)→R(G) which forgets the Real structure is injective.
2. Any complex G-representationV which is a Real representation can only possess a unique Real structure up to isomorphisms of Real G-representations.
Proof . Letρ:R(G)→RR(G) be the map [V]7→
V ⊕ι∗GV , σV⊕ι∗ GV
,
where σV⊕ι∗
GV(u, w) = (w, u). Let [(V, σV)]∈RR(G). Then ρ◦i([(V, σV)]) =
V ⊕ι∗GV , σV⊕ι∗ GV
.
We claim that [(V⊕V, σV ⊕σV)] = [(V⊕ι∗GV , σV⊕ι∗
GV)], which is easily seen to be true because of the Real G-representation isomorphism
f : V ⊕ι∗GV →V ⊕V,
(u, w)7→(u+σV(w), i(u−σV(w))).
It follows that ρ◦iamounts to multiplication by 2 onRR(G), and is therefore injective because RR(G) is a free abelian group generated by irreducible Real representations. Henceiis injective.
(2) is simply a restatement of (1).
Proposition 2.17 makes it legitimate to regard RR(G) as a subring of R(G). From now on we view [V]∈R(G) as an element inRR(G) ifV possesses a compatible Real structure.
Proposition 2.18. Let G be a compact Real Lie group. LetV be an irreducible complex repre- sentation of G. Then
1. [V]∈RR(G,R) iff there exists a G-invariant symmetric nondegenerate bilinear form B : V ×σ∗GV →C.
2. [V ⊕V]∈RR(G,H) iff there exists aG-invariant skew-symmetric nondegenerate bilinear form B:V ×σG∗V →C.
3. [V ⊕σ∗GV] ∈ RR(G,C) iff there does not exist any G-invariant nondegenerate bilinear form on V ×σ∗GV.
Proof . By Proposition 2.16, V is a Real representation of real type iff there exists f ∈ Hom(V, σ∗GV)G such thatf2 = IdV. One can define a bilinear form B :V ×σG∗V →Cby
B(v1, v2) =hv1, f(v2)i, (2.3)
where h,i is a G-invariant Hermitian inner product on V. It can be easily seen that B is G-invariant, symmetric and non-degenerate. Conversely, given aG-invariant symmetric non- degenerate bilinear form onV×σ∗GV and using equation (2.3), we can definef ∈Hom(V, σG∗V)G, which squares to identity. Part (2) and (3) follow similarly.
Proposition2.18 leads to the following
Definition 2.19. Let Gbe a compact Real Lie group. LetV be an irreducible complex repre- sentation ofG. Define, with respect toσG,
1. V to be of real type if there exists a G-invariant symmetric nondegenerate bilinear form B :V ×σ∗GV →C.
2. V to be of quaternionic type if there exists a G-invariant skew-symmetric nondegenerate bilinear form B :V ×σG∗V →C.
3. V to be of complex type if V σG∗V.
The abelian group generated by classes of irreducible complex representation of typeFis denoted by R(G,F).
Definition 2.20. If V is a complex G-representation equipped with an anti-linear endomor- phismJV such thatJV(gv) =σG(g)J(v) andJ2 =−IdV, then we sayV is aQuaternionic repre- sentation ofG. LetRepH(G) be the category of Quaternionic representations ofG. A morphism betweenV andW ∈ Rep
H(G) is a linear transformation fromV toW which commutes with G and respectJV andJW. We denote Mor(V, W) by HomG(V, W)(JV,JW). An irreducible Quater- nionic representation is an irreducible object in RepH(G, σG). The Quaternionic representation group of G, denoted by RH(G), is the Grothendieck group of Rep
H(G). Let RH(G,F) be the abelian group generated by the isomorphism classes of irreducible Quaternionic representations with Fas the commuting field.
Remark 2.21. The tensor product of two Quaternionic representations V and W is a Real representation as JV ⊗JW is an anti-linear involution which is compatible with σG. Similarly the tensor product of a Real representation and a Quaternionic representation is a Quaternionic representation. To put it succinctly,RR(G)⊕RH(G) is aZ2-graded ring, withRR(G) being the degree 0 piece andRH(G) the degree −1 piece. Later on we will assignRH(G) with a different degree so as to be compatible with the description of the coefficient ring KR∗G(pt).
Proposition 2.22. RH(G), as an abelian group, is generated by the following
1. [V ⊕σ∗GV] where V is an irreducible complex representation of real type with J(u, w) = (−w, u). Its commuting field is H.
2. [V⊕σG∗V], whereV is an irreducible complex representation of complex type withJ(u, w) = (−w, u). Its commuting field is C.
3. [V], where V is an irreducible complex representation of quaternionic type. Its commuting field is R.
Proof . The proof proceeds in the same fashion as does the proof for Proposition 2.16.
Corollary 2.23.
1. We have that RR(G,R) ∼=RH(G,H) ∼=R(G,R), RH(G,R) ∼=RR(G,H) ∼=R(G,H) and RR(G,C)∼=RH(G,C) as abelian groups.
2. RR(G) is isomorphic to RH(G) as abelian groups.
Proof . The result follows easily from Proposition2.16, Definition2.19and Proposition2.22.
Proposition 2.24.
1. The map j:RH(G)→R(G) which forgets the Quaternionic structure is injective.
2. Any complex G-representation which is a Quaternionic representation can only possess a unique Quaternionic structure up to isomorphisms of Quaternionic G-representations.
Proof . The proof proceeds in the same fashion as does the proof for Proposition2.17. It suffices to show that, ifη:R(G)→RH(G) is the map [V]7→[(V ⊕σG∗V , σV⊕ι∗
GV)], thenη◦j amounts to multiplication by 2, i.e. (V⊕σ∗GV , σV⊕σ∗
GV)∼= (V⊕V, J⊕J), whereσV⊕σ∗
GV(u, w) = (−w, u).
This is true because of the QuaternionicG-representation isomorphism f : V ⊕ι∗GV →V ⊕V,
(u, w)7→(u+J w, i(u−J w)).
Example 2.25. We shall illustrate the similarities and differences of the various aforemen- tioned representation groups with an example. Let G = Q8 ×C3, the direct product of the quaternion group and the cyclic group of order 3, equipped with the trivial involution. There are 5 irreducible complex representations of Q8, namely, the 4 1-dimensional representations which become trivial on restriction to the center Z of Q8 and descend to the 4 1-dimensional representations of Q8/Z ∼=Z2 ⊕Z2, and the 2-dimensional faithful representation. We denote these representations by 1Q8, ρ(0,1), ρ(1,0), ρ(1,1) and ρQ respectively. Similarly, we let 1C3, ρζ and ρζ2 be the three 1-dimensional complex irreducible representations of C3. It can be easily seen that
R(Q8,R) =Z·[1Q8]⊕Z·[ρ(0,1)]⊕Z·[ρ(1,0)]⊕Z·[ρ(1,1)], R(Q8,C) = 0,
R(Q8,H) =Z·[ρQ], R(C3,R) =Z·[1C3],
R(C3,C) =Z·[ρζ]⊕Z·[ρζ2], R(C3,H) = 0.
It follows that
R(G,R) = M
x∈{1Q
8,ρ(0,1),ρ(1,0),ρ(1,1)}
Z·[x⊗1b C3]∼=Z4,
R(G,C) = M
x∈{1Q8,ρ(0,1),ρ(1,0),ρ(1,1),ρQ} y∈{ρζ,ρζ2}
Z·[x⊗y]b ∼=Z10,
R(G,H) =Z·[ρQ⊗1b C3]∼=Z, RR(G,R) = M
x∈{1Q8,ρ(0,1),ρ(1,0),ρ(1,1)}
Z·[x⊗1b C3]∼=Z4,
RR(G,C) = M
x∈{1Q
8,ρ(0,1),ρ(1,0),ρ(1,1),ρQ}
Z·[x⊗ρb ζ⊕x⊗ρb ζ2]∼=Z5, RR(G,H) =Z·[ρQ⊗1b C3 ⊕ρQ⊗1b C3]∼=Z,
RH(G,R) =Z·[ρQ⊗1b C3]∼=Z, RH(G,C) = M
x∈{1Q
8,ρ(0,1),ρ(1,0),ρ(1,1),ρQ}
Z·[x⊗ρb ζ⊕x⊗ρb ζ2]∼=Z5, RH(G,H) = M
x∈{1Q8,ρ(0,1),ρ(1,0),ρ(1,1)}
Z·[x⊗1b C3 ⊕x⊗1b C3]∼=Z4.
Some representations above should be equipped with suitable Real or Quaternionic structures given in Propositions 2.16 and 2.22. For example, the Real structure of ρQ⊗1b C3 ⊕ρQ⊗1b C3 in RR(G,H) is given by swapping the two coordinates.
2.3 KR-theory
KR-theory was first introduced by Atiyah in [2] and used to derive the 8-periodicity of KO- theory from the 2-periodicity of complex K-theory. KR-theory was motivated by the index theory of real elliptic operators.
Definition 2.26.
1. AReal space is a pair (X, σX) whereX is a topological space equipped with an involutive homeomorphism σX, i.e. σX2 = IdX. We will sometimes suppress the notation σX and simply use X to denote the Real space, if there is no danger of confusion about the involutive homeomorphism. A Real pair is a pair (X, Y) where Y is a closed subspace of X invariant underσX.
2. Let Rp,q be the Euclidean space Rp+q equipped with the involution which is identity on the first q coordinates and negation on the lastp-coordinates. Let Bp,q and Sp,q be the unit ball and sphere inRp,q with the inherited involution.
3. A Real vector bundle (to be distinguished from the usual real vector bundle) over X is a complex vector bundle E over X which itself is also a Real space with involutive homeomorphism σE satisfying
(a) σX◦p=p◦σE, wherep:E →X is the projection map, (b) σE maps Ex toEσX(x) anti-linearly.
A Quaternionic vector bundle (to be distinguished from the usual quaternionic vector bundle) over X is a complex vector bundle E overX equipped with an anti-linear liftσE
of σX such thatσE2 =−IdE.
4. Let X be a Real space. The ring KR(X) is the Grothendieck group of the isomorphism classes of Real vector bundles overX, equipped with the usual product structure induced by tensor product of vector bundles over C. The relative KR-theory for a Real pair KR(X, Y) can be similarly defined. In general, the graded KR-theory ring of the Real pair (X, Y) is given by
KR∗(X, Y) :=
7
M
q=0
KR−q(X, Y),
where
KR−q(X, Y) :=KR X×B0,q, X×S0,q∪Y ×B0,q .
The ring structure ofKR∗is extended from that ofKR, in a way analogous to the case of complexK-theory. The number of graded pieces, which is 8, is a result of Bott periodicity forKR-theory (cf. [2]).
Note that whenσX = IdX, then KR(X)∼=KO(X). On the other hand, if X×Z2 is given the involution which swaps the two copies of X, then KR(X×Z2) ∼= K(X). Also, if X is equipped with the trivial involution, then KR(X×S2,0) ∼=KSC(X), the Grothendieck group of homotopy classes of self-conjugate bundles overX (cf. [2]). In this way, it is natural to view KR-theory as a unifying thread of KO-theory,K-theory andKSC-theory.
On top of the Real structure, we may further add compatible group actions and define equivariantKR-theory.
Definition 2.27.
1. A Real G-space X is a quadruple (X, G, σX, σG) where a groupGacts onX andσG is an involutive automorphism of Gsuch that
σX(g·x) =σG(g)·σX(x).
2. A Real G-vector bundle E over a RealG-spaceX is a Real vector bundle and aG-bundle overX, and it is also a Real G-space.
3. In a similar spirit, one can define equivariantKR-theory KR∗G(X, Y). Notice that theG- actions onB0,q and S0,q in the definition ofKR−qG (X, Y) are trivial.
Definition 2.28.
1. Let K∗(+) be the complex K-theory of a point extended to a Z8-graded algebra over K0(pt) ∼= Z, i.e. K∗(+) ∼= Z[β]
β4−1 . Here β ∈ K−2(+) is the class of the reduced canonical bundle onCP1 ∼=S2.
2. Let σX∗ be the map defined on (equivariant) vector bundles on X by σX∗E := σX∗E. The involution induced by σX∗ onKG∗(X) is also denoted by σX∗ for simplicity.
In the following proposition, we collect, for reader’s convenience, some basic results of KR- theory (cf. [15, Section 2]), some of which are stated in the more general context of equivariant KR-theory.
Proposition 2.29.
1. We have
KR∗(pt)∼=Z[η, µ]
2η, η3, µη, µ2−4 ,
where η∈KR−1(pt),µ∈KR−4(pt)represents the reduced Hopf bundles of RP1 and HP1 respectively.
2. Let c:KR∗G(X)→KG∗(X) be the homomorphism which forgets the Real structure of Real vector bundles, and r : KG∗(X) → KR∗G(X) be the realification map defined by [E] 7→
[E⊕σG∗σX∗E]. Then we have the following relations
(a) c(1) = 1, c(η) = 0,c(µ) = 2β2, where β ∈K−2(pt) is the Bott class, (b) r(1) = 2, r(β) =η2, r(β2) =µ, r(β3) = 0,
(c) r(xc(y)) = r(x)y, cr(x) = x+σG∗σX∗x and rc(y) = 2y for x ∈ KG∗(X) and y ∈ KR∗G(X), where KG∗(X) is extended to a Z8-graded algebra by Bott periodicity.
Proof . (1) is given in [15, Section 2]. The proof of (2) is the same as in the nonequivariant
case, which is given in [2].
Definition 2.30. A Quaternionic G-vector bundle over a Real space X is a complex vector bundleEequipped with an anti-linear vector bundle endomorphismJonEsuch thatJ2 =−IdE and J(g·v) = σG(g)·J(v). Let KHG∗(X) be the corresponding K-theory constructed using QuaternionicG-bundles over X.
By generalizing the discussion preceding Lemma 5.2 in [15] to the equivariant and graded setting, we define a natural transformation
t: KHG−q(X)→KRG−q−4(X) which sends
0−→E1 −→f E2−→0 to
0−→π∗(H⊗CE1)−→g π∗(H⊗CE2)−→0, where
1) Ei, i = 1,2 are equivariant Quaternionic vector bundles on X×R0,q equipped with the Quaternionic structures JEi,
2) f is an equivariant Quaternionic vector bundle homomorphism which is an isomorphism outside X× {0},
3) π :X×R0,q+4 →X×R0,q is the projection map,
4) H⊗CEi is the equivariant Real vector bundles equipped with the Real structureJ⊗JEi, 5) g is an equivariant Real vector bundle homomorphism defined by g(v, w⊗e) = (v, vw⊗
f(e)).
One can easily show by generalizing the discussion in the last section of [5] that Proposition 2.31. t is an isomorphism.
2.4 The module structure of KR-theory of compact simply-connected Lie groups
The following structure theorem for KR-theory, due to Seymour, is crucial in his computation of KR∗(pt)-module structure ofKR∗(G).
Theorem 2.32([15, Theorem 4.2]). Suppose thatK∗(X)is a free abelian group and decomposed by the involution σX∗ into the following summands
K∗(X) =M+⊕M−⊕T⊕σ∗XT,
where σ∗X is identity onM+ and negation on M−. Suppose further that there existh1, . . . , hn∈ KR∗(X) such that c(h1), . . . , c(hn) form a basis of the K∗(+)-module K∗(+)⊗(M+⊕M−).
Then, as KR∗(pt)-modules,
KR∗(X)∼=F ⊕r(K∗(+)⊗T),
where F is the free KR∗(pt)-module generated by h1, . . . , hn.
Remark 2.33. IfT = 0, then the conditions in Theorem2.32are equivalent toK∗(X) being free abelian andc:KR∗(X)→K∗(X) being surjective. In this special case the theorem implies that the map KR∗(X)⊗KR∗(pt)K∗(pt)→K∗(X) defined by a⊗b7→c(a)·bis a ring isomorphism.
This smacks of the definition of weakly equivariant formality (cf. Remark 2.8) and inspires us to define a similar notion for equivariant KR-theory (cf. Definition4.2). We say a real space is real formal if it satisfies the conditions of Theorem2.32.
Definition 2.34. LetσRbe the complex conjugation ofU(n) orU(∞), andσHbe the symplectic type involution g7→JmgJm−1 on U(2m), orU(2∞).
For any Real spaceX,KR−1(X) is isomorphic to the abelian group of equivariant homotopy classes of maps from X to U(∞) which respect σX and σR on U(∞). Similarly, KR−5(X), which is isomorphic to KH−1(X) by Proposition 2.31, is isomorphic to the abelian group of equivariant homotopy classes of maps from X to U(2∞) which respect σX and σH on U(2∞) (cf. remarks in the last two paragraphs of Appendix of [15]). We can define maps analogous to those in Definition 2.1in the context ofKR-theory.
Definition 2.35. Let δR : RR(G) → KR−1(G) and δH : RH(G) → KR−5(G) be group ho- momorphisms which send a Real (resp. Quaternionic) representation to the KR-theory element represented by its homotopy class.
Proposition 2.36. If ρ ∈ RR(G), then δR(ρ) is represented by the complex of vector bundles in Proposition 2.2 equipped with the Real structure given by
ι: G×R×V →G×R×V, (g, t, v)7→(σG(g), t, v).
If ρ ∈RH(G), then δH(ρ) can be similarly represented, with the Real structure replaced by the Quaternionic structure.
From this point on until the end of this section, we further assume that G is connected and simply-connected unless otherwise specified. It is known thatR(G) is a polynomial ring over Z generated by fundamental representations, which are permuted byσ∗G(cf. [15, Lemma 5.5]). Let
R(G)∼=Z
ϕ1, . . . , ϕr, θ1, . . . , θs, γ1, . . . , γt, σG∗γ1, . . . , σ∗Gγt
,
where ϕi ∈RR(G,R), θj ∈ RH(G,R), γk ∈R(G,C). Then K∗(G), as a free abelian group, is generated by square-free monomials inδ(ϕ1), . . ., δ(ϕr),δ(θ1), . . ., δ(θs),δ(γ1), . . ., δ(γt),δ(σG∗γ1), . . ., δ(σG∗γt). Using Theorem2.32, Seymour obtained
Theorem 2.37 ([15, Theorem 5.6]).
1. Suppose that σG∗ acts as identity onR(G), i.e. any irreducible Real representation of G is either of real type or quaternionic type. Then as KR∗(pt)-modules,
KR∗(G)∼=∧KR∗(pt)(δR(ϕ1), . . . , δR(ϕr), δH(θ1), . . . , δH(θs)).
2. More generally, c(δR(ϕi)) = δ(ϕi), c(δH(θj)) = β2 ·δ(θj), and there exist λ1, . . . , λt ∈ KR0(G) such that c(λk) =β3·δ(γk)δ(σG∗γk), and
KR∗(G)∼=P ⊕T·P as KR∗(pt)-module, where
• P ∼=V
KR∗(pt)(δR(ϕ1), . . . , δR(ϕr), δH(θ1), . . . , δH(θs), λ1, . . . , λt),
• T is the additive abelian group generated by the set {r(βi·δ(γ1)ε1· · ·δ(γt)εtδ(σ∗Gγ1)ν1· · ·δ(σ∗Gγt)νt)},
where ε1, . . . , εt, ν1, . . . , νt are either 0or 1, εk andνk are not equal to 1 at the same time for 1≤k≤t, and the first index k0 where εk0 = 1 is less than the first index k1 where νk1 = 1.
Moreover,
(a) λ2k= 0 for all 1≤k≤t,
(b) δR(ϕi)2 and δH(θj)2 are divisible by η.
Definition 2.38. Let ωt:=δt,1−νt and
ri,ε1,...,εt,ν1,...,νt :=r βi·δ(γ1)ε1· · ·δ(γt)εtδ(σG∗(γ1))ν1· · ·δ σG∗(γt)νt
∈T.
Corollary 2.39.
1. KR∗(G)is generated byδR(ϕ1), . . ., δR(ϕr),δH(θ1), . . ., δH(θs),λ1, . . ., λtandri,ε1,...,εt,ν1,...,νt
∈T as an algebra over KR∗(pt).
2.
ri,ε2 1,...,εt,ν1,...,νt =
η2λω11· · ·λωtt, if ri,ε1,...,εt,ν1,...,νt is of degree −1 or −5,
±µλω11· · ·λωtt, if ri,ε1,...,εt,ν1,...,νt is of degree −2 or −6,
0 otherwise.
The sign depends oni, ε1, . . . , εt,ν1, . . . , νt and can be determined using formulae from(2) of Proposition 2.29.
3. ri,ε1,...,εt,ν1,...,νtη= 0, and ri,ε1,...,εt,ν1,...,νtµ= 2ri+2,ε1,...,εt,ν1,...,νt.
Proof . The Corollary follows easily from the various properties of the realification map and the complexification map in Proposition 2.29, and the fact thatc(λk) =β3·δ(γk)δ(σG∗γk). For example,
ri,ε1,...,εt,ν1,...,νtη=r βi·δ(γ1)ε1· · ·δ(γt)εtδ(σ∗G(γ1))ν1· · ·δ σ∗G(γt)νt
c(η)
= 0, ri,ε1,...,εt,ν1,...,νtµ=r βi·δ(γ1)ε1· · ·δ(γt)εtδ(σG∗(γ1))ν1· · ·δ σ∗G(γt)νt
c(µ)
=ri+2,ε1,...,εt,ν1,...,νt.
In fact Theorems 2.4 and 2.32 also yield the following description of module structure of KR-theory of a compact connected Real Lie group with torsion-free fundamental group with a restriction on the types of the Real representations.
Theorem 2.40. Let G be a compact connected real Lie group with π1(G) torsion-free. Sup- pose that R(G,C) = 0, i.e. σ∗G acts as identity on R(G). Then KR∗(G) is isomorphic to
∧∗KR∗(pt)(Im(eδR),Im(eδH)) as KR∗(pt)-modules.
As we see from Theorem2.37and Corollary2.39, to get a full description of the ring structure of KR∗(G), it remains to figure out δR(ϕi)2 and δH(θj)2. We will, in the end, obtain formulae for the squares by way of computing the ring structure of KR∗G(G) and applying the forgetful map. In particular, we will show that δR(ϕi)2 and δH(θj)2 in general are non-zero. So, unlike the complex K-theory,KR∗(G) is not an exterior algebra in general. Nevertheless, KR∗(G) is not far from being an exterior algebra, in the sense of the following
Corollary 2.41.
1. KR∗(pt)2, which is the ring obtained by inverting the prime 2 in KR∗(pt), is isomorphic to Z1
2, µ
/(µ2−4)∼=Z1
2, β2
/((β2)2−1).
2. Suppose that R(G,C) = 0. KR∗(G)2, which is the ring obtained by inverting the prime 2 in KR∗(G), is isomorphic to, asKR∗(pt)2-algebra
^
KR∗(pt)2
δR(ϕ1), . . . , δR(ϕr), δH(θ1), . . . , δH(θs) .
3 The coef f icient ring KR
∗G(pt)
In this section, we assume that Gis a compact Real Lie group, and will prove a result on the coefficient ring KR∗G(pt). In [5], all graded pieces of KR∗G(pt) were worked out using Real Clifford G-modules. We record them in the following
Proposition 3.1. KR−qG (pt), as abelian groups, for 0 ≤ q ≤ 7, are isomorphic to RR(G), RR(G)/ρ(R(G)), R(G)/j(RH(G)), 0, RH(G), RH(G)/η(R(G)), R(G)/i(RR(G)) and 0 re- spectively, where the maps i, j, ρ, η are as in Propositions 2.17 and2.24.
Remark 3.2. Note from the above proposition thatKR0G(pt)⊕KR−4G (pt)∼=RR(G)⊕RH(G).
In this way we can view RR(G)⊕RH(G) as a graded ring where RR(G) is of degree 0 and RH(G) of degree−4.
Proposition 3.3.
1. Suppose R(G,C) = 0. Then the map
f : (RR(G,R)⊕RH(G,R))⊗KR∗(pt)→KR∗G(pt), ρ1⊗x1⊕ρ2⊗x2 7→ρ1·x1+ρ2·x2
is an isomorphism of graded rings.
2. In general,
f : (RR(G,R)⊕RH(G,R))⊗KR∗(pt)⊕r(R(G,C)⊗K∗(+))→KR∗G(pt), ρ1⊗x1⊕ρ2⊗x2⊕r(ρ3⊗βi)7→ρ1·x1+ρ2·x2+r(ρ3·βi)
is an isomorphism of graded abelian groups.
3. If ρ is an irreducible complex representation of complex type, then ηr(βi ·ρ) = 0 and µr(βi·ρ) = 2r(βi+2·ρ).
Proof . The proposition follows by verifying the isomorphism in different degree pieces against the description in Proposition3.1. For example, in degree 0,
RR(G,R)⊗KR0(pt)⊕RH(G,R)⊗KR−4(pt)⊕r(R(G,C)⊗K0(+))
=RR(G,R)⊕RH(G,R)⊗Zµ⊕RR(G,C)
∼=RR(G,R)⊕RR(G,H)⊕RR(G,C)
(if [V]∈RH(G,R), then [V]·µ= [V ⊕V]∈RR(G,H))
=RR(G) =KR0G(pt).
(3) follows from Proposition 2.29.
Remark 3.4. In [5], KRG∗(X), where the G-action is trivial, is given as the following direct sum of abelian groups
RR(G,R)⊗KR∗(X)⊕RR(G,C)⊗KC∗(X)⊕RR(G,H)⊗KH∗(X),
whereKC∗(X) andKH∗(X) are Grothendieck groups of the so-called ‘Complex vector bundles’
and ‘Quaternionic vector bundles’ of X. We find Proposition 3.3, which is motivated by this description, better because the ring structure of the coefficient ring is more apparent when cast in this light. The proposition is, as we will see in the next section, a consequence of a structure theorem of equivariant KR-theory (Theorem 4.5), and therefore still holds true if the point is replaced by any general space X with trivial G-action.
4 Equivariant KR-theory rings
of compact simply-connected Lie groups
Throughout this section we assume that Gis a compact, connected and simply-connected Real Lie group unless otherwise specified. We will prove the main result of this paper, Theorem4.33, which gives the ring structure of KR∗G(G). Our strategy is outlined as follows.
1. We obtain a result on the structure of KR∗G(G) (Corollary 4.10) which is analogous to Theorem 2.37 and Proposition 3.3. We define δRG(ϕi), δHG(θj), λGk and rρ,i,εG 1,...,εt,ν1,...,νt (cf. Definition 4.8and Corollary4.11), which generate KR∗G(G) as aKR∗G(pt)-algebra, as a result of Corollary4.10. We show that (λGk)2 = 0 (cf. Proposition 4.13).
2. We compute the module structure ofKR∗(U(n),σ
F)(U(n), σF) forF=R andH.
3. LetT be the maximal torus of diagonal matrices inU(n) and, by abuse of notation,σRbe the inversion map onT,σH be the involution on U(n)/T (wheren= 2m is even) defined by gT 7→JmgT. We show that the restriction map
p∗G : KR∗(U(n),σ
R)(U(n), σR)→KR∗(T,σ
R)(T, σR) and the map
qG∗ : KR∗(U(2m),σ
H)(U(2m), σH)→KR∗(U(2m),σ
H)(U(2m)/T×T, σH×σR)
induced by the Weyl covering map qG : U(2m)/T ×T → U(2m), (gT, t) 7→ gtg−1, are injective.
4. Letσnbe the class of the standard representation ofU(n). We pass the computation of the two squares δG
R(σn)2 ∈ KR∗(U(n),σ
R)(U(n), σR) and δG
H(σ2m)2 ∈ KR(U(2m),σ∗
H)(U(2m), σH) through the induced mapp∗G and qG∗ to their images and get equations (4.2) and (4.3) in Proposition4.29.
5. Applying induced maps ϕ∗i and θj∗ to equations (4.2) and (4.3) yields equations (4.4) and (4.5) in Theorem 4.30 which, together with Proposition 4.13 and some relations among η, µ and rρ,i,εG 1,...,εt,ν1,...,νt deduced from Proposition 2.29, describe completely the ring structure of KR∗G(G) (cf. Theorem 4.33).
Remark 4.1.
1. Seymour first suggested the analogues of Steps 3, 4 and 5 in the ordinaryKR-theory case in [15] in an attempt to computeδR(ϕi)2 andδH(θj)2, but failed to establish Step 3, which he assumed to be true to make conjectures aboutδR(ϕi)2.
2. In equivariant complex K-theory,KG∗(G/T ×T) ∼=KT∗(T) for any compact Lie group G, and the two mapsp∗G(the restriction map induced by the inclusionT ,→G) andq∗Gwhich is induced by the Weyl covering map are the same. If π1(G) is torsion-free, then these two maps are shown to be injective (cf. [9]. In fact it is even shown there that the maps inject onto the Weyl invariants of KT∗(T)). In the case of equivariant KR-theory, things are more complicated. First of all, while in the case where (G, σG) = (U(n), σR), it is true that KR∗G(G/T ×T) ∼=KR∗T(T), and p∗G and q∗G are the same, it is no longer true in the case where (G, σG) = (U(2m), σH). In Step 3, we use q∗G for the quaternionic type involution case because we find that it admits an easier description thanp∗Gdoes. Second, we do not know whether p∗G and q∗G are injective for general compact Real Lie groups (equipped with any Lie group involution). For our purpose it is sufficient to show the injectivity results in Step 3.
4.1 A structure theorem
Definition 4.2. A G-space X is aReal equivariantly formal space if 1) Gis a compact Real Lie group,
2) X is a weakly equivariantly formal G-space, and
3) the forgetful map KRG∗(X)→KR∗(X) admits a sectionsR:KR∗(X)→KR∗G(X) which is aKR∗(pt)-module homomorphism.
Remark 4.3. IfX is a weakly equivariantly formalG-space, then the forgetful map KG∗(X)→ K∗(X) admits a (not necessarily unique) section s:K∗(X) →KG∗(X) which is a group homo- morphism.
Definition 4.4. For a section s : K∗(X) → KG∗(X) (resp. sR : KR∗(X) → KR∗G(X)) and a ∈K∗(X) (resp. a∈ KR∗(X)), we call s(a) (resp. sR(a)) a (Real) equivariant lift of a, with respect tos(resp. sR).
We first prove a structure theorem of equivariant KR-theory of Real equivariantly formal spaces.
Theorem 4.5. Let X be a Real equivariantly formal space. For any elementa∈K∗(X) (resp.
a∈KR∗(X)), letaG∈KG∗(X) (resp.aG∈KR∗G(X))be a(Real)equivariant lift ofawith respect to a group homomorphic section s(resp.sR which is aKR∗(pt)-module homomorphism). Then the map
f : (RR(G,R)⊕RH(G,R))⊗KR∗(X)⊕r(R(G,C)⊗K∗(X))→KR∗G(X), ρ1⊗a1⊕r(ρ2⊗a2)7→ρ1·(a1)G⊕r(ρ2·(a2)G).
